Docker is exactly like OpenVZ. It became popular because they really emphasize their OpenVZ Application Templates feature, and made it much more user friendly.
So users of Docker, instead of following this guide: https://openvz.org/Application_Templates
They write a Dockerfile, which in a simple case might be:
COPY index.html /usr/share/nginx/html
Then you can push your nice website container to the public registry, ssh to your machine and pull it from the registry. Of course you can have your own private registry (we do) so you can have proprietary docker containers that run your apps/sites.
From my perspective, the answer to your question would be: Always prefer Docker over OpenVZ, they are the same technology but Docker is easier to use.
But I've never really invested in OpenVZ so maybe there's some feature that Docker doesn't have.
They are built around the same set of kernel features and as far as I can see offer exactly the same abstractions.
For example Phusion Baseimage is a Docker baseimage that's similar to an OpenVZ container in the sense that it emulates a full running Ubuntu environment. It has is uses but the Docker community rather sees containers that encapsulate a single application with no external processes.
Docker does that, too. Actually, I'm running docker containers as a fast and easy replacement for VirtualBox VMs.
I had to change /etc config files, mange where logfiles went and how they were handled, all of this is described on a diffing filesystem.
I might be misinterpreting what you mean though. I have no experience with OpenVZ and little with Docker.